Research Article Existence of Positive Solutions for m-point Boundary Value Problems on Time Scales

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1 Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 29, Article ID , 12 pages doi:1.1155/29/ Research Article Existence of Positive Solutions for m-point Boundary Value Problems on Time Scales Ying Zhang and ShiDong Qiao Department of Mathematics, Shanxi Datong University, Datong, Shanxi 379, China Correspondence should be addressed to Ying Zhang, Received 27 August 28; Revised 24 November 28; Accepted 14 January 29 Recommended by Binggen Zhang We study the one-dimensional p-laplacian m-point boundary value problem p u Δ t Δ a t f t, u t, t, 1 T, u, u 1 a iu ξ i,wheret is a time scale, p s s p 2 s, p > 1, some new results are obtained for the existence of at least one, two, and three positive solution/solutions of the above problem by using Krasnosel skll s fixed point theorem, new fixed point theorem due to Avery and Henderson, as well as Leggett-Williams fixed point theorem. This is probably the first time the existence of positive solutions of one-dimensional p-laplacian m- point boundary value problem on time scales has been studied. Copyright q 29 Y. Zhang and S. Qiao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction With the development of p-laplacian dynamic equations and theory of time scales, a few authors focused their interest on the study of boundary value problems for p-laplacian dynamic equations on time scales. The readers are referred to the paper 1 7. In 25, He 1 considered the following boundary value problems: p u Δ t a t f u t, t,t T, u B u Δ η, u Δ T or 1.1 u Δ, u T B 1 u Δ η, where T is a time scales, p s s p 2 s, p > 1, η,ρ t T. The author showed the existence of at least two positive solutions by way of a new double fixed point theorem.

2 2 Discrete Dynamics in Nature and Society In 24, Anderson et al. 2 used the virtue of the fixed point theorem of cone and obtained the existence of at least one solution of the boundary value problem: g u Δ t c t f u, a<t<b, u a B u Δ γ, u Δ b. 1.2 In 27, Geng and Zhu 3 used the Avery-Peterson and another fixed theorem of cone and obtained the existence of three positive solutions of the boundary value problem: p u Δ t a t f u t, t,t T, u B u Δ η, u Δ T. 1.3 Also, in 27, Sun and Li 4 discussed the existence of at least one, two or three positive solutions of the following boundary value problem: p u Δ t Δ h t f u σ t, t a, b T, u a B u Δ a, u Δ σ b. 1.4 In this paper, we are concerned with the existence of multiple positive solutions to the m-point boundary value problem for the one dimension p-laplcaian dynamic equation on time scale T p u Δ t Δ a t f t, u t, t, 1 T, u, u 1 a i u ξ i, 1.5 where T is a time scale, p s s p 2 s, p > 1, < ξ 1 < ξ 2 < < < 1, a i,i 1, 2,...,m 3, a >, and H 1 < 1; H 2 f C rd, 1 T,,, ; H 3 a C rd, 1 T,, and there exists t, 1 such that a t >. In this paper, we have organized the paper as follows. In Section 2, we give some lemmas which are needed later. In Section 3, we apply the Krassnoselskiifs 8 fixed point theorem to prove the existence of at least one positive solution to the MBVP 1.5.InSection 4, conditions for the existence of at least two positive solutions to the MBVP 1.5 are discussed by using Avery and Henderson 9 fixed point theorem. In Section 5, to prove the existence of at least three positive solutions to the MBVP 1.5 are discussed by using Leggett and Williams 1 fixed point theorem. For completeness, we introduce the following concepts and properties on time scales. A time scale T is a nonempty closed subset of R, assume that T has the topology that it inherits from the standards topology on R.

3 Discrete Dynamics in Nature and Society 3 Definition 1.1. Let T be a time scale, for t T, one defines the forward jump operator σ : T T by σ t inf{s T : s > t}, and the backward jump operator ρ : T T by ρ t sup{s T : s < t}, while the graininess function μ : T, is defined by μ t σ t t. Ifσ t >t,one says that is right-scattered, while if ρ t <t,one says that tis left-scattered. Also, if t<sup T and σ t t, then t is called right-dense, and if t>inf T and ρ t t, then t is called left-dense. One also needs below the set T k as follows: if T has a left-scattered maximum m, then T k T m, otherwise T k T. For instance, if sup T, then T k T. Definition 1.2. Assume f : T R is a function and let t T. Then, one defines f Δ t to be the number provided it exists with the property that any given ε>, there is a neighborhood U of t such that [ f σ t f s ] f Δ t σ t s ε σ t s, 1.6 for all s U. One says that f is delta differentiable or in short: differentiable on T provided f Δ t exist for all t T. If T R, then f Δ t f t, if T Z, then f Δ t Δf t. A function f : T R. i If f is continuous, then f is rd-continuous. ii The jump operator σ is rd-continuous. iii If f is rd-continuous, then so is f σ. A function F : T R is called an antidervative of f : T R, provided F Δ t f t holds for all t T k. One defines the definite integral by b a f t Δt F b F a. 1.7 For all a, b T.Iff Δ t, then f is nondecreasing. 2. The Preliminary Lemmas Lemma 2.1 see 5, 6. Assume that H 1 H 3 hold. Then u t is a solution of the MBVP 1.5 on, 1 T if and only if t u t t t s q a i a τ f τ, u τ a, 2.1 where q s s q 2 s, 1/p 1/q 1, and q>1.

4 4 Discrete Dynamics in Nature and Society Lemma 2.2. Assume that conditions H 1 H 3 are satisfied, then the solution of the MBVP 1.5 on, 1 T satisfies u t, t, 1 T. 2.2 Lemma 2.3 see 5. If the conditions H 1 H 3 are satisfied, then u t γ u, t [, 1 ], 2.3 where u sup u t, t,1 T { } a 1 γ min,a,ξ 1. 1 a 2.4 Lemma 2.4 see 6. min t ξ,1 Au t min{au 1,Au }. Let E denote the Banach space C rd, 1 T with the norm u sup t,1 T u t. Define the cone P E, by P { u E u t, t [, 1 ] min u t γ u, uis concave }. 2.5 The solutions of MBVP 1.5 are the points of the operator A defined by t Au t t t s q a i a τ f τ, u τ a u t. 2.6 So, AP P. It is easy to check that A : P P is completely continuous. 3. Existence of at least One Positive Solutions Theorem 3.1 see 8. Let E be a Banach space, and let P E be a cone. Assume Ω 1 and Ω 2 are open boundary subsets of E with Ω 1, Ω 1 Ω 2, and let A : P Ω 2 \ Ω 1 P be a completely continuous operator such that either i Au u for u P Ω 1, Au u for u P Ω 2 ; or ii Au u for u P Ω 1, Au u for u P Ω 2 hold. Then A has a fixed point in P Ω 2 \ Ω 1.

5 Discrete Dynamics in Nature and Society 5 Theorem 3.2. Assume conditions H 1 H 3 are satisfied. In addition, suppose there exist numbers < r < R < such that f t, u p m p r, if t,σ 1, u r, and f t, u p Mγ p R, if t, 1, R u<, where M γ q s a τ Δτ Δs, m a τ Δτ Δs. 3.1 Then the MBVP 1.5 has at least one positive solution. Proof. Define the cone P as in 2.5, define a completely continuous integral operator A : P P by t Au t t t s q a i a τ f τ, u τ a. 3.2 From H 1 H 3, Lemmas 2.1 and 2.2, AP P.Ifu P with u r, then we get t Au t t s q a i a τ f τ, u τ t t q p m p r rm a τ Δτ Δs a τ Δτ Δs a r u. 3.3 This implies that Au u. So, if we set Ω 1 {u C rd, 1 u <r}, then Au u, for u P Ω 1. Let us now set Ω 2 {u C rd, 1 u <R}.

6 6 Discrete Dynamics in Nature and Society Then for u P with u < R, by Lemma 2.4 we have u t γ u, t, 1. Therefore, we have Au t Au ξ ξ q a τ f τ, u τ a i ξ q s 1 ξ a i ξ i q a τ f τ, u τ Δτ s q ξ ξ q s ξ q s q p Mγ p R 1 q s a τ Δτ Δs MγR a q a τ Δτ Δs a τ f τ, u τ Δs u. 3.4 Hence, Au u for u P Ω 2. Thus by the Theorem 3.1, A has a fixed point u in P Ω 2 \ Ω 1. Therefore, the MBVP 1.5 has at least one positive solution. 4. Existence of at least Two Positive Solutions In this section, we apply the Avery-Henderson fixed point theorem 9 to prove the existence of at least two positive solutions to the nonlinear MBVP 1.5. Theorem 4.1 see Avery and Henderson 9. Let P be a cone in a real Banach space E. Set P Φ,ρ 3 { u P Φ u <ρ3 }. 4.1

7 Discrete Dynamics in Nature and Society 7 If ν and Φ are increasing, nonnegative continuous functionals on P, let θ be a nonnegative continuous functional on P with θ such that, for some positive constants ρ 3 and M>, Φ u θ u ν u and u MΦ u, for all u P Φ,ρ 3. Suppose that there exist positive numbers ρ 1 <ρ 2 <ρ 3 such that θ λu λθ u for all λ 1andu P θ, ρ 2. If A : P Φ,ρ 3 P is a completely continuous operator satisfying i Φ Au >ρ 3 for all u P Φ,ρ 3 ; ii θ Au <ρ 2 for all u P θ, ρ 2 ; iii P ν, ρ 1 / φ and ν Au >ρ 1 for all u P ν, ρ 1, then A has at least two fixed points u 1 and u 2 such that ρ 1 <ν u 1 with θ u 1 <ρ 2 and ρ 3 < u 2 with Φ u 2 <ρ 3. Let l, 1 T and < <l<1. Define the increasing, nonnegative and continuous functionals Φ, θ,and ν on P, by Φ u u,θ u u, and ν u u l. From Lemma 2.4, for each u P, Φ u θ u ν u. In addition, for each u P, Lemma 2.3 implies Φ u u γ u. Thus, u < 1 Φ u, u P. 4.2 γ We also see that θ andθ λu λθ u for all λ 1andu P θ, q. Theorem 4.2. Assume H 1 H 3 hold, suppose there exist positive numbers ρ 1 <ρ 2 <ρ 3, such that the function f satisfies the following conditions: B 1 f t, u > p mγ p ρ 1, for t,l and u γρ 1,ρ 1 ; B 2 f t, u < p m p ρ 2, for t, 1 and u,ρ 2 ; B 3 f t, u > p Mγ p ρ 3, for t,l and u ρ 3, 1/γ ρ 3. Then the MBVP 1.5 has at least two positive solutions u 1 and u 2 such that u 1 t >ρ 1 with u 1 l <ρ 2 and u 2 l >ρ 2 with u 2 l <ρ 3. Proof. We now verify that all of the conditions of Theorem 4.1 are satisfied. Define the cone P as 2.5, define a completely continuous integral operator A : P P by t Au t t t s q a i a τ f τ, u τ a. 4.3 M and m as in 3.1. To verify that condition i of Theorem 4.1 holds, we choose u P Φ,ρ 3, then Φ u ρ 3. This implies ρ 3 u 1/γ Φ u. Notethat u 1/γ Φ u

8 8 Discrete Dynamics in Nature and Society 1/γ ρ 3. We have ρ 3 u t 1/γ ρ 3, for t, 1 T. As a consequence of B 3, f t, u > p Mγ p ρ 3, for t,l T. Since Au P, we have from Lemma 2.2, Φ Au Au ξ q a τ f τ, u τ a i ξ q s 1 ξ a i ξ i q 1 a τ f τ, u τ a i ξ i s q a τ f τ, u τ ξ q s ξ q s ξ q p Mγ q s a τ Δτ Δs p ρ3 Mγρ 3 a q a τ Δτ Δs ρ Then condition i of Theorem 4.1 holds. Let u P θ, ρ 2. Then θ u ρ 2. This implies u t u 1/γ ρ 2, for t, 1. From B 2, we have θ Au Au 4.5 mρ 2 a τ Δτ Δs a ρ 2 u. Hence condition ii of Theorem 4.1 holds.

9 Discrete Dynamics in Nature and Society 9 If we first define u t ρ 1 /2, for t, 1 T, then ν u ρ 1 /2 <ρ 1. So P ν, ρ 1 / φ. Now, let u P ν, ρ 1, then ν u u l ρ 1. This mean that ρ 1 /γ u t u ρ 1. From B 1 and Lemma 2.4,weget ν Au Au l Au ξ q a τ f τ, u τ a i ξ q s 1 ξ a i ξ i q a τ f τ, u τ Δτ Δs s q a τ f τ, u τ ξ q s ξ ξ q s ξ q p mγ q s a τ Δτ Δs p ρ1 mγρ1 a q a τ Δτ Δs ρ 1. Then condition iii of Theorem 4.1 holds. Since all conditions of Theorem 4.1 are satisfied, the MBVP 1.5 has at least two positive solutions u 1 and u 2 such that u 1 t > ρ 1 with u 1 l < ρ 2 and u 2 l > ρ 2 with u 2 l <ρ Existence of at least Three Positive Solutions We will use the Leggett-Williams fixed point theorem 1 to prove the existence of at least three positive solutions to the nonlinear MBVP 1.5. Theorem 5.1 see Leggett and Williams 1. Let P be a cone in the real Banach space E. Set Pr { x P x <r }, P Ψ,a,b { x P a Ψ x, x b }. 5.1

10 1 Discrete Dynamics in Nature and Society Suppose A : Pr Pr be a completely continuous operator and be a nonnegative continuous concave functional on P with Ψ u u for all u Pr. If there exists <ρ 1 < ρ 2 < 1/γ ρ 2 ρ 3 such that the following condition hold: i {u P Ψ,ρ 2, 1/γ ρ 2 Ψ u >ρ 2 } / φ and Ψ Au >ρ 2 for all u P Ψ,ρ 2, 1/γ ρ 2 ; ii Au <ρ 1 for u ρ 1 ; iii Ψ Au >ρ 2 for u P Ψ,ρ 2, 1/γ ρ 2 with Au > 1/γ ρ 2, then A has at least three fixed points u 1,u 2 and u 3 in Pr satisfying u 1 <ρ 1, Ψ u 2 >ρ 2,ρ 1 < u 3 with Ψ u 2 <ρ 2. Theorem 5.2. Assume H 1 H 3 hold. Suppose that there exist constants <ρ 1 <ρ 2 < 1/γ ρ 2 ρ 3 such that C 1 f t, u p m p ρ 3, for t,l and u,ρ 3 ; C 2 f t, u > p Mγ p ρ 2, for t,l and u ρ 2, 1/γ ρ 2 ; C 3 f t, u < p m p ρ 1, for t, 1 and u,ρ 1. Then the MBVP 1.5 has at least three positive solutions u 1,u 2, and u 3 such that u 1 ξ <ρ 1,u 2 l >ρ 2,u 3 ξ >ρ 1 with u 3 l <ρ 2. Proof. The conditions of Theorem 5.1 will be shown to be satisfied. Define the nonnegative continuous concave functional Ψ : P, to be Ψ u u, the cone P as in 2.5, M and m as in 3.1. We have Ψ u u for all u P. If u P ρ3, then u ρ 3, and from assumption C 1, then we have t Au t t s q a i a τ f τ, u τ t t q p m p ρ3 mρ 3 a τ Δτ Δs a τ Δτ Δs a ρ This implies that Au ρ 3. Thus, we have A : P ρ3 P ρ3. Since 1/γ ρ 2 P Ψ,ρ 2, 1/γ ρ 2 and Ψ 1/γ ρ 2 1/γ ρ 2 > ρ 2, {u P Ψ,ρ 2, 1/γ ρ 2 Ψ u > ρ 2 } / φ.

11 Discrete Dynamics in Nature and Society 11 For u P Ψ,ρ 2, 1/γ ρ 2 we have ρ 2 u u 1/γ ρ 2. Using assumption C 2, f t, u > p Mγ p ρ 2, we obtain Ψ Au Au ξ q a τ f τ, u τ a i ξ q s 1 ξ a a i ξ i q a τ f τ, u τ s q a τ f τ, u τ ξ q s ξ ξ q s q p Mγ p ρ2 q s a τ Δτ Δs Mγρ 1 2 a q a τ Δτ Δs ρ Hence, condition i of Theorem 5.1 holds. If u ρ 1, from assumption C 3,weobtain Au t q p m p ρ1 a τ Δτ Δs mρ 1 a τ Δτ Δs a ρ This implies that Au ρ 1. Consequently, condition ii of Theorem 5.1 holds.

12 12 Discrete Dynamics in Nature and Society We suppose that u P Ψ,ρ 2,ρ 3, with Au > 1/γ ρ 2. Then we get Ψ Au Au ξ ξ q s ξ q p Mγ q s a τ Δτ Δs p ρ2 Mγρ 1 2 a q a τ Δτ Δs ρ Hence, condition iii of Theorem 5.1 holds. Because all of the hypotheses of the Leggett-Williams fixed point theorem are satisfied, the nonlinear MBVP 1.5 has at least three positive solutions u 1,u 2, and u 3 such that u 1 ξ < ρ 1,u 2 l >ρ 2, and u 3 ξ >ρ 1 with u 3 l <ρ 2. Acknowledgment This work is supported by the Research and Development Foundation of College of Shanxi Province no References 1 Z. He, Double positive solutions of three-point boundary value problems for p-laplacian dynamic equations on time scales, Journal of Computational and Applied Mathematics, vol. 182, no. 2, pp , D. R. Anderson, R. I. Avery, and J. Henderson, Existence of solutions for a one dimensional p- Laplacian on time scales, Journal of Difference Equations and Applications, vol. 1, no. 1, pp , F. Geng and D. Zhu, Multiple results of p-laplacian dynamic equations on time scales, Applied Mathematics and Computation, vol. 193, no. 2, pp , H.-R. Sun and W.-T. Li, Existence theory for positive solutions to one-dimensional p-laplacian boundary value problems on time scales, Journal of Differential Equations, vol. 24, no. 2, pp , R. Ma, Positive solutions of a nonlinear m-point boundary value problem, Computers & Mathematics with Applications, vol. 42, no. 6-7, pp , C. Bai and J. Fang, Existence of multiple positive solutions for nonlinear m-point boundary value problems, Journal of Mathematical Analysis and Applications, vol. 281, no. 1, pp , W.-T. Li and H.-R. Sun, Multiple positive solutions for nonlinear dynamical systems on a measure chain, Journal of Computational and Applied Mathematics, vol. 162, no. 2, pp , D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, R. I. Avery and J. Henderson, Two positive fixed points of nonlinear operators on ordered Banach spaces, Communications on Applied Nonlinear Analysis, vol. 8, no. 1, pp , R. W. Leggett and L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana University Mathematics Journal, vol. 28, no. 4, pp , 1979.